in \(\mathbb {P}_0^{(n)}\) -probability, where the rate is nearly minimax optimal under the \(L_2\)  norm (cf. Theorem 2.8 in [1]}). We defer the formal statement to Lemma REF . Therefore, by verifying the regularity conditions in Theorem REF , we conclude that with the same choice of \(\lambda \) , the posterior distribution will contract at \(f_0\)  at the same nearly minimax optimal rate under the \(L_2\)  norm.
